%% This should be typeset with TeXShop after selecting TeX and DVI from the Typeset menu.

\magnification = 2000 

\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros

\hsize 6.5 true in
\vsize 9.25 true in
\hoffset = 0.00 true in
\voffset -0.5 true in
\parskip=3pt


\vglue -20pt

% File Name as ATO: z --> conj(z) + aa z^2.pdf

\cl{ {\bf Complex Map $ z\mapsto \hbox{\rm conj}(z) + aa\cdot z^2$}
 \footnote{*}{\verysmall 
 This file is from the 3D-XplorMath project.  Please see: \hfill\break
   \phantom{http://} http://3D-XplorMath.org/}} 
%\cl{\bf (Default: $z\to z^2+2z$)}
\noindent
Look at other functions, e.g. $z\to z^2$, $ \exp$, and their ATOs first.
\lf
The map $ z\mapsto \hbox{\rm conj}(z) + aa\cdot z^2$ is of course a map from
the complex plane to itsself. The harmless looking ``conj'' is responsible for the
fact that this map is not complex differentiable and therefore not a ``conformal''
map, that is, a map for which the angles between any two curves and their
images are the same. It is clearly visible in the image that the squares of the 
domain grid are mapped to rectangles and even to parallelograms in the range.
\vskip0mm\noindent
\hbox{         \hskip-0.4cm
{\vbox{ \hsize=0.5\hsize   % \phantom{.} \vskip-1.5cm
                           \includegraphics[width=1.3in]{nonConformal.png} }
  \vbox{ \hsize=0.5\hsize   \ni
  					The image also shows two ``fold lines''. We
					observe \break
					that interior points of the domain are 
					map\-ped so that they lie on the
					boundary  of the image.  For a complex 
					dif\-ferentiable  function this can never
					happen  by the \lf
					\Ph{0.5}{\it Open Mapping Theorem} . 
					See the default morph. \lf
					\rightline{H. K. }       
   } 
   }
}


\bye